Question

In: Statistics and Probability

A botanist measures the sepal widths (in cm) of 150 iris flowers, 50 of type iris...

A botanist measures the sepal widths (in cm) of 150 iris flowers, 50 of type iris setosa, 50 of type iris virginica, and 50 of type iris versicolor. The data is noted below. Of interest is whether sepal width differs, on average, among the three iris types. You may assume that, for each iris type, the sample may be treated as a simple random sample.

a. What are the null and alternative hypotheses?

b. What is the value of the test statistic?

c. What is the approximate p-value?

d. State and verify (using plots or descriptive statistics) the additional two assumptions required for the p-value in c) to be valid.

e. Using a significance level of = 0.05, state your conclusions in the language of the problem.

Sepal Width and Class = 3.5 Iris-setosa 3 Iris-setosa 3.2 Iris-setosa 3.1 Iris-setosa 3.6 Iris-setosa 3.9 Iris-setosa 3.4 Iris-setosa 3.4 Iris-setosa 2.9 Iris-setosa 3.1 Iris-setosa 3.7 Iris-setosa 3.4 Iris-setosa 3 Iris-setosa 3 Iris-setosa 4 Iris-setosa 4.4 Iris-setosa 3.9 Iris-setosa 3.5 Iris-setosa 3.8 Iris-setosa 3.8 Iris-setosa 3.4 Iris-setosa 3.7 Iris-setosa 3.6 Iris-setosa 3.3 Iris-setosa 3.4 Iris-setosa 3 Iris-setosa 3.4 Iris-setosa 3.5 Iris-setosa 3.4 Iris-setosa 3.2 Iris-setosa 3.1 Iris-setosa 3.4 Iris-setosa 4.1 Iris-setosa 4.2 Iris-setosa 3.1 Iris-setosa 3.2 Iris-setosa 3.5 Iris-setosa 3.1 Iris-setosa 3 Iris-setosa 3.4 Iris-setosa 3.5 Iris-setosa 2.3 Iris-setosa 3.2 Iris-setosa 3.5 Iris-setosa 3.8 Iris-setosa 3 Iris-setosa 3.8 Iris-setosa 3.2 Iris-setosa 3.7 Iris-setosa 3.3 Iris-setosa 3.2 Iris-versicolor 3.2 Iris-versicolor 3.1 Iris-versicolor 2.3 Iris-versicolor 2.8 Iris-versicolor 2.8 Iris-versicolor 3.3 Iris-versicolor 2.4 Iris-versicolor 2.9 Iris-versicolor 2.7 Iris-versicolor 2 Iris-versicolor 3 Iris-versicolor 2.2 Iris-versicolor 2.9 Iris-versicolor 2.9 Iris-versicolor 3.1 Iris-versicolor 3 Iris-versicolor 2.7 Iris-versicolor 2.2 Iris-versicolor 2.5 Iris-versicolor 3.2 Iris-versicolor 2.8 Iris-versicolor 2.5 Iris-versicolor 2.8 Iris-versicolor 2.9 Iris-versicolor 3 Iris-versicolor 2.8 Iris-versicolor 3 Iris-versicolor 2.9 Iris-versicolor 2.6 Iris-versicolor 2.4 Iris-versicolor 2.4 Iris-versicolor 2.7 Iris-versicolor 2.7 Iris-versicolor 3 Iris-versicolor 3.4 Iris-versicolor 3.1 Iris-versicolor 2.3 Iris-versicolor 3 Iris-versicolor 2.5 Iris-versicolor 2.6 Iris-versicolor 3 Iris-versicolor 2.6 Iris-versicolor 2.3 Iris-versicolor 2.7 Iris-versicolor 3 Iris-versicolor 2.9 Iris-versicolor 2.9 Iris-versicolor 2.5 Iris-versicolor 2.8 Iris-versicolor 3.3 Iris-virginica 2.7 Iris-virginica 3 Iris-virginica 2.9 Iris-virginica 3 Iris-virginica 3 Iris-virginica 2.5 Iris-virginica 2.9 Iris-virginica 2.5 Iris-virginica 3.6 Iris-virginica 3.2 Iris-virginica 2.7 Iris-virginica 3 Iris-virginica 2.5 Iris-virginica 2.8 Iris-virginica 3.2 Iris-virginica 3 Iris-virginica 3.8 Iris-virginica 2.6 Iris-virginica 2.2 Iris-virginica 3.2 Iris-virginica 2.8 Iris-virginica 2.8 Iris-virginica 2.7 Iris-virginica 3.3 Iris-virginica 3.2 Iris-virginica 2.8 Iris-virginica 3 Iris-virginica 2.8 Iris-virginica 3 Iris-virginica 2.8 Iris-virginica 3.8 Iris-virginica 2.8 Iris-virginica 2.8 Iris-virginica 2.6 Iris-virginica 3 Iris-virginica 3.4 Iris-virginica 3.1 Iris-virginica 3 Iris-virginica 3.1 Iris-virginica 3.1 Iris-virginica 3.1 Iris-virginica 2.7 Iris-virginica 3.2 Iris-virginica 3.3 Iris-virginica 3 Iris-virginica 2.5 Iris-virginica 3 Iris-virginica 3.4 Iris-virginica 3 Iris-virginica

Expert Solution

Given data set in tabular form:

Species	SepalWidthCm
Iris-setosa	3.5
Iris-setosa	3
Iris-setosa	3.2
Iris-setosa	3.1
Iris-setosa	3.6
Iris-setosa	3.9
Iris-setosa	3.4
Iris-setosa	3.4
Iris-setosa	2.9
Iris-setosa	3.1
Iris-setosa	3.7
Iris-setosa	3.4
Iris-setosa	3
Iris-setosa	3
Iris-setosa	4
Iris-setosa	4.4
Iris-setosa	3.9
Iris-setosa	3.5
Iris-setosa	3.8
Iris-setosa	3.8
Iris-setosa	3.4
Iris-setosa	3.7
Iris-setosa	3.6
Iris-setosa	3.3
Iris-setosa	3.4
Iris-setosa	3
Iris-setosa	3.4
Iris-setosa	3.5
Iris-setosa	3.4
Iris-setosa	3.2
Iris-setosa	3.1
Iris-setosa	3.4
Iris-setosa	4.1
Iris-setosa	4.2
Iris-setosa	3.1
Iris-setosa	3.2
Iris-setosa	3.5
Iris-setosa	3.1
Iris-setosa	3
Iris-setosa	3.4
Iris-setosa	3.5
Iris-setosa	2.3
Iris-setosa	3.2
Iris-setosa	3.5
Iris-setosa	3.8
Iris-setosa	3
Iris-setosa	3.8
Iris-setosa	3.2
Iris-setosa	3.7
Iris-setosa	3.3
Iris-versicolor	3.2
Iris-versicolor	3.2
Iris-versicolor	3.1
Iris-versicolor	2.3
Iris-versicolor	2.8
Iris-versicolor	2.8
Iris-versicolor	3.3
Iris-versicolor	2.4
Iris-versicolor	2.9
Iris-versicolor	2.7
Iris-versicolor	2
Iris-versicolor	3
Iris-versicolor	2.2
Iris-versicolor	2.9
Iris-versicolor	2.9
Iris-versicolor	3.1
Iris-versicolor	3
Iris-versicolor	2.7
Iris-versicolor	2.2
Iris-versicolor	2.5
Iris-versicolor	3.2
Iris-versicolor	2.8
Iris-versicolor	2.5
Iris-versicolor	2.8
Iris-versicolor	2.9
Iris-versicolor	3
Iris-versicolor	2.8
Iris-versicolor	3
Iris-versicolor	2.9
Iris-versicolor	2.6
Iris-versicolor	2.4
Iris-versicolor	2.4
Iris-versicolor	2.7
Iris-versicolor	2.7
Iris-versicolor	3
Iris-versicolor	3.4
Iris-versicolor	3.1
Iris-versicolor	2.3
Iris-versicolor	3
Iris-versicolor	2.5
Iris-versicolor	2.6
Iris-versicolor	3
Iris-versicolor	2.6
Iris-versicolor	2.3
Iris-versicolor	2.7
Iris-versicolor	3
Iris-versicolor	2.9
Iris-versicolor	2.9
Iris-versicolor	2.5
Iris-versicolor	2.8
Iris-virginica	3.3
Iris-virginica	2.7
Iris-virginica	3
Iris-virginica	2.9
Iris-virginica	3
Iris-virginica	3
Iris-virginica	2.5
Iris-virginica	2.9
Iris-virginica	2.5
Iris-virginica	3.6
Iris-virginica	3.2
Iris-virginica	2.7
Iris-virginica	3
Iris-virginica	2.5
Iris-virginica	2.8
Iris-virginica	3.2
Iris-virginica	3
Iris-virginica	3.8
Iris-virginica	2.6
Iris-virginica	2.2
Iris-virginica	3.2
Iris-virginica	2.8
Iris-virginica	2.8
Iris-virginica	2.7
Iris-virginica	3.3
Iris-virginica	3.2
Iris-virginica	2.8
Iris-virginica	3
Iris-virginica	2.8
Iris-virginica	3
Iris-virginica	2.8
Iris-virginica	3.8
Iris-virginica	2.8
Iris-virginica	2.8
Iris-virginica	2.6
Iris-virginica	3
Iris-virginica	3.4
Iris-virginica	3.1
Iris-virginica	3
Iris-virginica	3.1
Iris-virginica	3.1
Iris-virginica	3.1
Iris-virginica	2.7
Iris-virginica	3.2
Iris-virginica	3.3
Iris-virginica	3
Iris-virginica	2.5
Iris-virginica	3
Iris-virginica	3.4
Iris-virginica	3

Now, finding the mean of widht for 3 types.

_s = 3.418

_ve = 2.77

_vi = 2.872

1.)

Null Hyptheses:

Ho: On an average the means of all three groups are significanlty same.

_s = _ve = _vi

Alternate Hypotheses:

Ha: On an average the means of all three groups are significanlty different.

_s_ve_vi

2.)

To check the above Hypotheses, we need to perform One way ANOVA and calculate F statistic.

The Summary table of ANOVA is given below for alpha = 0.05 and degree of freedom = 3-1 = 2:

Summary of Data
	Treatments
	Setosa	Verisoclor	Virginica	Total
N	50	50	50	150
?X	170.9	138.5	148.7	458.1
Mean	3.418	2.77	2.974	3.054
?X²	591.25	388.47	447.33	1427.05
Std.Dev.	0.381	0.3138	0.3225	0.4336

Result details are given below:

Result Details
Source	SS	df	MS
Between-treatments	10.9776	2	5.4888	F = 47.36446
Within-treatments	17.035	147	0.1159
Total	28.0126	149

F-statistic Value = 47.36

3.) p-value for above test statistic is:

f-ratio value = 47.36

p = 0.0001

4.) There are 2 assumptions to check:

-> whether data for 3 categories follows normal distribution.

-> standard deviation for 3 categories are equal.

Checking the first one:

To check the normal distribution we will check for Histograms of 3 categories.

-> Setosa:

From the above graph we can see that there are some outliers in setosa, but graph is almost normal.

-> Versiocolor

Although not perfect, but data for Versicolor is also near to normal state.

-> Virginica:

For virginica the data looks normal.

We can ssume that data for 3 types follows normal distribution approximately and continue with ANOVA test.

Checking second Assumption:

The standrad devition of 3 groups are:

0.38, 0.31, 0.32

These are almost same. So our second assumotion is also satisfied.

5.) given alpha = 0.05

our p-value = 0.0001

since p < 0.05

We can reject Null Hypotheses and say that on an average mean sepal width of 3 categories are statistically different.

orchestra answered 2 years ago

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A botanist measures the sepal widths (in cm) of 150 iris flowers, 50 of type iris...

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